Recent works have characterized the function-space inductive bias of infinite-width bounded-norm single-hidden-layer neural networks as a kind of bounded-variation-type space. This novel neural network Banach space encompasses many classical multivariate function spaces, including certain Sobolev spaces and the spectral Barron spaces. Notably, this Banach space also includes functions that exhibit less classical regularity, such as those that only vary in a few directions. On bounded domains, it is well-established that the Gaussian reproducing kernel Hilbert space (RKHS) strictly embeds into this Banach space, demonstrating a clear gap between the Gaussian RKHS and the neural network Banach space. It turns out that when investigating these spaces on unbounded domains, e.g., all of $\mathbb{R}^d$, the story is fundamentally different. We establish the following fundamental result: Certain functions that lie in the Gaussian RKHS have infinite norm in the neural network Banach space. This provides a nontrivial gap between kernel methods and neural networks by exhibiting functions that kernel methods easily represent, whereas neural networks cannot.
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎电信学院，LTCI实验室）
Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with infinite dimensional output spaces unlocks many more applications, such as functional regression, structured output prediction, and structured data representation. However, these sophisticated schemes crucially rely on the kernel trick in the output space, so that most of previous…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in the Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional Hilbert space, convergence in the 2-Sinkhorn divergence is {\it strictly weaker} than convergence in the exact 2-Wasserstein distance. Specifically, a sequence of centered Gaussian measures converges in the 2-Sinkhorn divergence if the corresponding covariance operators converge in the Hilbert-Schmidt norm. This is in contrast to the previous known result that a sequence of c…
Given a reproducing kernel Hilbert space H of real-valued functions and a suitable measure mu over the source space D (subset of R), we decompose H as the sum of a subspace of centered functions for mu and its orthogonal in H. This decomposition leads to a special case of ANOVA kernels, for which the functional ANOVA representation of the best predictor can be elegantly derived, either in an interpolation or regularization framework. The proposed kernels appear to be particularly convenient for analyzing the e ffect of each (group of) variable(s) and computing sensitivity indices without recur…
Kernel methods are powerful tools to capture nonlinear patterns behind data. They implicitly learn high (even infinite) dimensional nonlinear features in the Reproducing Kernel Hilbert Space (RKHS) while making the computation tractable by leveraging the kernel trick. Classic kernel methods learn a single layer of nonlinear features, whose representational power may be limited. Motivated by recent success of deep neural networks (DNNs) that learn multi-layer hierarchical representations, we propose a Stacked Kernel Network (SKN) that learns a hierarchy of RKHS-based nonlinear features. SKN int…
CREST, ENSAE, IP Paris（巴黎理工学院国立统计与经济管理学院经济与统计研究中心）；The University of Tokyo（东京大学）；INRIA, Université Paris-Saclay（巴黎萨克雷大学法国国家信息与自动化研究所）；Riken AIP（理化学研究所先进智能项目中心）
Gradient Langevin dynamics (GLD) and stochastic GLD (SGLD) have attracted considerable attention lately, as a way to provide convergence guarantees in a non-convex setting. However, the known rates grow exponentially with the dimension of the space. In this work, we provide a convergence analysis of GLD and SGLD when the optimization space is an infinite dimensional Hilbert space. More precisely, we derive non-asymptotic, dimension-free convergence rates for GLD/SGLD when performing regularized non-convex optimization in a reproducing kernel Hilbert space. Amongst others, the convergence analy…
Based on the theory of reproducing kernel Hilbert space (RKHS) and semiparametric method, we propose a new approach to nonlinear dimension reduction. The method extends the semiparametric method into a more generalized domain where both the interested parameters and nuisance parameters to be infinite dimensional. By casting the nonlinear dimensional reduction problem in a generalized semiparametric framework, we calculate the orthogonal complement space of generalized nuisance tangent space to derive the estimating equation. Solving the estimating equation by the theory of RKHS and regularizat…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Tübingen, Germany（德国图宾根）
This short technical report presents some learning theory results on vector-valued reproducing kernel Hilbert space (RKHS) regression, where the input space is allowed to be non-compact and the output space is a (possibly infinite-dimensional) Hilbert space. Our approach is based on the integral operator technique using spectral theory for non-compact operators. We place a particular emphasis on obtaining results with as few assumptions as possible; as such we only use Chebyshev's inequality, and no effort is made to obtain the best rates or constants.
INRIA（法国国家信息与自动化研究所）；Département d’Informatique de l’École Normale Supérieure（巴黎高等师范学院计算机系）；PSL Research University（巴黎文理研究大学）；London School of Economics（伦敦政治经济学院）
Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard way (for example at all points of an interval) for rich function classes is a notoriously challenging problem. We propose a unified and modular convex optimization framework, relying on second-order cone (SOC) tightening, to encode hard affine SDP constraints on function derivatives, for models belonging to vector-valued reproducing kernel Hilbert spaces (v…
Suppose that $Y$ is a scalar and $X$ is a second-order stochastic process, where $Y$ and $X$ are conditionally independent given the random variables $ξ_1,...,ξ_p$ which belong to the closed span $L_X^2$ of $X$. This paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of $L_X^2$ with the reproducing kernel Hilbert space of $X$ provides a platform for a seamless extension from the finite- to infinite-dimensional settings. It also facilitates convenient computational algorithms that can be applied to a variety of mode…
Columbia University（哥伦比亚大学）；Pennsylvania State University（宾夕法尼亚州立大学）
The main contribution of the paper is to show that Gaussian sketching of a kernel-Gram matrix $\boldsymbol K$ yields an operator whose counterpart in an RKHS $\mathcal H$, is a \emph{random projection} operator---in the spirit of Johnson-Lindenstrauss (J-L) lemma. To be precise, given a random matrix $Z$ with i.i.d. Gaussian entries, we show that a sketch $Z\boldsymbol{K}$ corresponds to a particular random operator in (infinite-dimensional) Hilbert space $\mathcal H$ that maps functions $f \in \mathcal H$ to a low-dimensional space $\mathbb R^d$, while preserving a weighted RKHS inner-product…
We generalize the orthonormal basis for the Gaussian RKHS described in \cite{MinhGaussian2010} to an infinite, continuously parametrized, family of orthonormal bases, along with some implications. The proofs are direct generalizations of those in \cite{MinhGaussian2010}.
Off-policy estimation for long-horizon problems is important in many real-life applications such as healthcare and robotics, where high-fidelity simulators may not be available and on-policy evaluation is expensive or impossible. Recently, \cite{liu18breaking} proposed an approach that avoids the \emph{curse of horizon} suffered by typical importance-sampling-based methods. While showing promising results, this approach is limited in practice as it requires data be drawn from the \emph{stationary distribution} of a \emph{known} behavior policy. In this work, we propose a novel approach that el…
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院LTCI实验室）
This paper investigates a novel algorithmic approach to data representation based on kernel methods. Assuming that the observations lie in a Hilbert space X, the introduced Kernel Autoencoder (KAE) is the composition of mappings from vector-valued Reproducing Kernel Hilbert Spaces (vv-RKHSs) that minimizes the expected reconstruction error. Beyond a first extension of the autoencoding scheme to possibly infinite dimensional Hilbert spaces, KAE further allows to autoencode any kind of data by choosing X to be itself a RKHS. A theoretical analysis of the model is carried out, providing a general…
Weighted log-rank tests are arguably the most widely used tests by practitioners for the two-sample problem in the context of right-censored data. Many approaches have been considered to make weighted log-rank tests more robust against a broader family of alternatives, among them, considering linear combinations of weighted log-rank tests, and taking the maximum among a finite collection of them. In this paper, we propose as test statistic the supremum of a collection of (potentially infinite) weight-indexed log-rank tests where the index space is the unit ball in a reproducing kernel Hilbert …
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
In this work, we present formulations for regularized Kullback-Leibler and Rényi divergences via the Alpha Log-Determinant (Log-Det) divergences between positive Hilbert-Schmidt operators on Hilbert spaces in two different settings, namely (i) covariance operators and Gaussian measures defined on reproducing kernel Hilbert spaces (RKHS); and (ii) Gaussian processes with squared integrable sample paths. For characteristic kernels, the first setting leads to divergences between arbitrary Borel probability measures on a complete, separable metric space. We show that the Alpha Log-Det divergences …
Machine learning has witnessed tremendous success in solving tasks depending on a single hyperparameter. When considering simultaneously a finite number of tasks, multi-task learning enables one to account for the similarities of the tasks via appropriate regularizers. A step further consists of learning a continuum of tasks for various loss functions. A promising approach, called \emph{Parametric Task Learning}, has paved the way in the continuum setting for affine models and piecewise-linear loss functions. In this work, we introduce a novel approach called \emph{Infinite Task Learning} whos…
This work presents a parametrized family of divergences, namely Alpha-Beta Log- Determinant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infinite-dimensional setting. The family of Alpha-Beta Log-Det divergences is highly general and contains many divergences as special cases, including the recently formulated infinite dimensional affine-invariant Riemannian distance and the infinite-dimensional Alpha Log-Det divergen…
Kernel ridge regression (KRR) is a well-known and popular nonparametric regression approach with many desirable properties, including minimax rate-optimality in estimating functions that belong to common reproducing kernel Hilbert spaces (RKHS). The approach, however, is computationally intensive for large data sets, due to the need to operate on a dense $n \times n$ kernel matrix, where $n$ is the sample size. Recently, various approximation schemes for solving KRR have been considered, and some analyzed. Some approaches such as Nyström approximation and sketching have been shown to preserve …
Dynamic Regret Reduces to Kernelized Static Regret
Università degli Studi di Milano Politecnico di Milano（米兰大学）；Bocconi University（博科尼大学）；King Abdullah University of Science and Technology (KAUST)（卡斯泰洛-阿卜杜勒-阿齐兹大学）
The problem of estimating the kernel mean in a reproducing kernel Hilbert space (RKHS) is central to kernel methods in that it is used by classical approaches (e.g., when centering a kernel PCA matrix), and it also forms the core inference step of modern kernel methods (e.g., kernel-based non-parametric tests) that rely on embedding probability distributions in RKHSs. Muandet et al. (2014) has shown that shrinkage can help in constructing "better" estimators of the kernel mean than the empirical estimator. The present paper studies the consistency and admissibility of the estimators in Muandet…
We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, using the Lévy-Khintchine formula, that any shift-invariant kernel given by a bounded, continuous and symmetric probability density function (pdf) of an infinitely divisible distribution on $\mathbb{R}^d$ is characteristic. We also present some closure property of such characteristi…
IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）；University of Washington（华盛顿大学）；University of Padova（帕多瓦大学）
Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random field with covariance proportional to the kernel associated with the RKHS. In this paper, we provide a statistical interpre…
University of California, Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）
Despite their many appealing properties, kernel methods are heavily affected by the curse of dimensionality. For instance, in the case of inner product kernels in $\mathbb{R}^d$, the Reproducing Kernel Hilbert Space (RKHS) norm is often very large for functions that depend strongly on a small subset of directions (ridge functions). Correspondingly, such functions are difficult to learn using kernel methods. This observation has motivated the study of generalizations of kernel methods, whereby the RKHS norm -- which is equivalent to a weighted $\ell_2$ norm -- is replaced by a weighted function…
We provide quantitative bounds measuring the $L^2$ difference in function space between the trajectory of a finite-width network trained on finitely many samples from the idealized kernel dynamics of infinite width and infinite data. An implication of the bounds is that the network is biased to learn the top eigenfunctions of the Neural Tangent Kernel not just on the training set but over the entire input space. This bias depends on the model architecture and input distribution alone and thus does not depend on the target function which does not need to be in the RKHS of the kernel. The result…
University of Washington（华盛顿大学）；Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Cornell University（康奈尔大学）；Carnegie Mellon University（卡内基梅隆大学）
This work introduces Bilinear Classes, a new structural framework, which permit generalization in reinforcement learning in a wide variety of settings through the use of function approximation. The framework incorporates nearly all existing models in which a polynomial sample complexity is achievable, and, notably, also includes new models, such as the Linear $Q^*/V^*$ model in which both the optimal $Q$-function and the optimal $V$-function are linear in some known feature space. Our main result provides an RL algorithm which has polynomial sample complexity for Bilinear Classes; notably, thi…
Reproducing kernel Hilbert spaces (RKHSs) are key elements of many non-parametric tools successfully used in signal processing, statistics, and machine learning. In this work, we aim to address three issues of the classical RKHS based techniques. First, they require the RKHS to be known a priori, which is unrealistic in many applications. Furthermore, the choice of RKHS affects the shape and smoothness of the solution, thus impacting its performance. Second, RKHSs are ill-equipped to deal with heterogeneous degrees of smoothness, i.e., with functions that are smooth in some parts of their doma…
University of Washington（华盛顿大学）；University College London（伦敦大学学院）；Carnegie Mellon University（卡内基梅隆大学）
Predictive State Representations (PSRs) are an expressive class of models for controlled stochastic processes. PSRs represent state as a set of predictions of future observable events. Because PSRs are defined entirely in terms of observable data, statistically consistent estimates of PSR parameters can be learned efficiently by manipulating moments of observed training data. Most learning algorithms for PSRs have assumed that actions and observations are finite with low cardinality. In this paper, we generalize PSRs to infinite sets of observations and actions, using the recent concept of Hil…
It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classic…
We study the estimation and prediction of functional autoregressive~(FAR) processes, a statistical tool for modeling functional time series data. Due to the infinite-dimensional nature of FAR processes, the existing literature addresses its inference via dimension reduction and theoretical results therein require the (unrealistic) assumption of fully observed functional time series. We propose an alternative inference framework based on Reproducing Kernel Hilbert Spaces~(RKHS). Specifically, a nuclear norm regularization method is proposed for estimating the transition operators of the FAR pro…
In this paper we propose a family of tractable kernels that is dense in the family of bounded positive semi-definite functions (i.e. can approximate any bounded kernel with arbitrary precision). We start by discussing the case of stationary kernels, and propose a family of spectral kernels that extends existing approaches such as spectral mixture kernels and sparse spectrum kernels. Our extension has two primary advantages. Firstly, unlike existing spectral approaches that yield infinite differentiability, the kernels we introduce allow learning the degree of differentiability of the latent fu…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能研究中心）
This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that these distances/divergences are fully represented by reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators associated with the corresponding covariance functions. Using this representation, we show that the Sinkhorn divergence between two centered Gaussian processes can be consistently and efficiently estimated from the divergence bet…
Reproducing Kernel Hilbert spaces (RKHS) have been a very successful tool in various areas of machine learning. Recently, Barron spaces have been used to prove bounds on the generalisation error for neural networks. Unfortunately, Barron spaces cannot be understood in terms of RKHS due to the strong nonlinear coupling of the weights. This can be solved by using the more general Reproducing Kernel Banach spaces (RKBS). We show that these Barron spaces belong to a class of integral RKBS. This class can also be understood as an infinite union of RKHS spaces. Furthermore, we show that the dual spa…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
This work studies the entropic regularization formulation of the 2-Wasserstein distance on an infinite-dimensional Hilbert space, in particular for the Gaussian setting. We first present the Minimum Mutual Information property, namely the joint measures of two Gaussian measures on Hilbert space with the smallest mutual information are joint Gaussian measures. This is the infinite-dimensional generalization of the Maximum Entropy property of Gaussian densities on Euclidean space. We then give closed form formulas for the optimal entropic transport plan, entropic 2-Wasserstein distance, and Sink…
We present a general nonlinear Bayesian filter for high-dimensional state estimation using the theory of reproducing kernel Hilbert space (RKHS). Applying kernel method and the representer theorem to perform linear quadratic estimation in a functional space, we derive a Bayesian recursive state estimator for a general nonlinear dynamical system in the original input space. Unlike existing nonlinear extensions of Kalman filter where the system dynamics are assumed known, the state-space representation for the Functional Bayesian Filter (FBF) is completely learned from measurement data in the fo…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
In this work we study two Riemannian distances between infinite-dimensional positive definite Hilbert-Schmidt operators, namely affine-invariant Riemannian and Log-Hilbert-Schmidt distances, in the context of covariance operators associated with functional stochastic processes, in particular Gaussian processes. Our first main results show that both distances converge in the Hilbert-Schmidt norm. Using concentration results for Hilbert space-valued random variables, we then show that both distances can be consistently and efficiently estimated from (i) sample covariance operators, (ii) finite, …
